可压流体方程及其耦合模型的定性性态

This project is devoted to the mathematical analysis of the qualitative behaviors of the solutions to compressible fluid-dynamical equations, including the existence, regularity and uniqueness of weak solutions to multi-dimensional compressible Navier-Stokes equations, the nonlinear stability of plane Couette and Poiseuille flow for multi-dimensional compressible Navier-Stokes equations and the Rayleigh-Taylor instability problem; the nonlinear stability of wave pattern, space-time asymptotical behaviors, and multi-scaled asymptotical limits of the local or global solutions to compressible Navier-Stokes(Euler)-Fokker-Planck equations, and the well-posedness of the initial boundary value problem for the coupled Navier-Stokes-Dacry/Porous models for two-phase flow motion; the well-posedness and asymptotical behaivors of the solutions to free boundary problem for multi-dimensional compressible (relativistic) Euler equations and the related models with the transport properties; the existence and nonlinear stability of boundary layer solutions to the mixture Boltzmann type equations, the contruction of Green’s function for multi-dimensional Vlasov-Poisson(Maxwell)-Boltzmann equations and the related coupling models, and the nonlinear stability and space-time behaviors of the basic waves profiles (such as shock profile, rarefaction wave) and their wave pattern for Vlasov-Poisson-Boltzmann(Fokker-Planck) equations.

本项目拟研究可压流体方程及其耦合模型的定性性态，包括高维可压(非等熵)Navier-Stokes方程弱解的存在性、正则性、唯一性；可压Navier-Stokes方程的Couette流稳定性与Rayleigh-Taylor不稳定性等；高维可压Navier-Stokes(Euler)-Fokker-Planck方程的非线性波现象、大时间行为、多尺度渐近极限等，高维Navier-Stokes-Dacry两相流耦合方程的适定性等；高维可压(相对论)Euler方程及相关模型的含真空自由边界问题的适定性、渐近行为等；高维Vlasov-Poisson(Maxwell)-Boltzmann方程等模型的格林函数构造、基本波的稳定性与时空逐点行为，混合气体的Boltzmann方程边界层问题及稳定性等。这些研究内容不仅是国际上十分重视的、有重要理论意义的、前沿主流课题，而且与工程应用紧密相关、有广泛的应用前景。

本项目执行以来的研究工作基本上按原计划执行，围绕“可压流体方程及其耦合模型的定性性态”，重点研究了可压缩Navier-Stokes方程及其耦合方程的适定性、渐近行为、时空逐点估计等，可压Navier-Stokes(Euler)-Vlasov-Fokker-Planck方程平面稀疏波的非线性稳定性、多尺度渐近极限等，高维可压缩(相对论)Euler方程及相关模型含真空自由边界问题的适定性、渐近行为等，高维Vlasov-Poisson-Boltzmann及相关模型的格林函数、谱结构、时空逐点行为等，高维Patlak-Keller-Segel方程及相关模型的适定性和渐近行为等，取得了多项进展。我们在国际重要数学刊物(比如Arch.Ration.Mech.Anal.,SIAM J.Math.Anal.等)发表或接受发表学术论文28篇，其中SCI论文24篇。